# Angle-Side-Angle (ASA) Congruence Theorem

Theorem: Two triangles are congruent if two angles and included side of one triangle are equal to the corresponding two angles and the included side of the other triangle.

**Given: ** In two triangles ABC and PQR, âˆ B = âˆ Q, âˆ C = âˆ R and BC = QR

**To Prove: ** Î”ABC â‰… Î”PQR

**Angle-Side-Angle (ASA) Congruence Theorem**

**Proof:**

**Case I : ** If AB = PQ, Î”ABC will be congruent to Î”PQR by the SAS criterion, and the theorem is proved.

**Case II : ** Suppose AB â‰ PQ and suppose AB is less than PQ. Take a point S on PQ such that QS = AB. Join RS

In Î” ABC and Î” SQR,

AB = SQ ... (Supposed)

BC = QR ... (Given)

âˆ B = âˆ Q ... (Given)

âˆ´ Î”ABC â‰… Î”SQR ... (SAS Criterion)

Hence, âˆ ACB = âˆ QRS ... (c.p.c.t.)

But âˆ ACB = âˆ QRP ... (Given)

âˆ´ âˆ QRP = âˆ QRS

which is impossible unless ray RS coincides with ray RP or S coincides with P.

âˆ´ AB must be equal to PQ.

**Case III : ** If we suppose that AB is greater than PQ, a similar argument applies and Î”ABC â‰… Î”PQR. Hence, in all cases, Î”ABC â‰… Î”PQR.